## General closed-form condition for enhanced transmission in subwavelength metallic gratings in both TE and TM polarizations |

Optics Express, Vol. 20, Issue 1, pp. 426-439 (2012)

http://dx.doi.org/10.1364/OE.20.000426

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### Abstract

We present an intuitive reasoning and derivation leading to an approximated, simple closed-form model for predicting and explaining the general emergence of enhanced transmission resonances through rectangular, optically thick metallic gratings in various configurations and polarizations. This model is based on an effective index approximation and it unifies in a simple way the underlying mechanism of enhanced transmission as emerging from standing wave resonances of the different diffraction orders of periodic structures. The model correctly predicts the conditions for the enhanced transmission resonances in various geometrical configurations, for both TE and TM polarizations, and in both the subwavelength and non-subwavelength spectral regimes, using the same underlying mechanism and one simple closed-form equation, and does not require explicitly invoking specific polarization dependent mechanisms. The known excitation of surface plasmons polaritons or slit cavity modes, emerge as limiting cases of a more general condition. This equation can be used to easily design and analyze the optical properties of a wide range of rectangular metallic transmission gratings.

© 2011 OSA

## 1. Introduction

1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**, 667–669 (1998). [CrossRef]

2. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science **297**, 820 –822 (2002). [CrossRef] [PubMed]

5. N. Livneh, A. Strauss, I. Schwarz, I. Rosenberg, A. Zimran, S. Yochelis, G. Chen, U. Banin, Y. Paltiel, and R. Rapaport, “Highly directional emission and photon beaming from nanocrystal quantum dots embedded in metallic nanoslit arrays,” Nano Lett. **11**, 1630–1635 (2011). [CrossRef] [PubMed]

6. F. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B **66** (2002). [CrossRef]

7. M. G. Harats, I. Schwarz, A. Zimran, U. Banin, G. Chen, and R. Rapaport, “Enhancement of two photon processes in quantum dots embedded in subwavelength metallic gratings,” Opt. Express **19**, 1617–1625 (2011). [CrossRef] [PubMed]

5. N. Livneh, A. Strauss, I. Schwarz, I. Rosenberg, A. Zimran, S. Yochelis, G. Chen, U. Banin, Y. Paltiel, and R. Rapaport, “Highly directional emission and photon beaming from nanocrystal quantum dots embedded in metallic nanoslit arrays,” Nano Lett. **11**, 1630–1635 (2011). [CrossRef] [PubMed]

8. X. Zhang, H. Liu, J. Tian, Y. Song, and L. Wang, “Band-Selective optical polarizer based on Gold-Nanowire plasmonic diffraction gratings,” Nano Lett. **8**, 2653–2658 (2008). [CrossRef] [PubMed]

9. F. Chien, C. Lin, J. Yih, K. Lee, C. Chang, P. Wei, C. Sun, and S. Chen, “Coupled waveguide-surface plasmon resonance biosensor with subwavelength grating,” Biosens. Bioelectron. **22**, 2737–2742 (2007). [CrossRef]

10. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. **83**, 2845 (1999). [CrossRef]

13. J. Shen and P. Platzman, “Properties of a one-dimensional metallophotonic crystal,” Phys. Rev. B **70** (2004). [CrossRef]

11. M. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B **66** (2002). [CrossRef]

14. K. G. Lee and Q. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. **95**, 103902 (2005). [CrossRef] [PubMed]

16. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science **305**, 847 –848 (2004). [CrossRef] [PubMed]

17. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. **82**, 729 (2010). [CrossRef]

6. F. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B **66** (2002). [CrossRef]

11. M. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B **66** (2002). [CrossRef]

15. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. **88**, 057403 (2002). [CrossRef] [PubMed]

20. A. V. Kats and A. Y. Nikitin, “Analytical treatment of anomalous transparency of a modulated metal film due to surface plasmon-polariton excitation,” Phys. Rev. B **70**, 235412 (2004). [CrossRef]

17. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. **82**, 729 (2010). [CrossRef]

11. M. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B **66** (2002). [CrossRef]

27. M. Moharam and T. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A **3**, 1780–1787 (1986). [CrossRef]

10. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. **83**, 2845 (1999). [CrossRef]

12. P. Lalanne, J. Hugonin, S. Astilean, M. Palamaru, and K. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. **2**, 48–51 (2000). [CrossRef]

15. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. **88**, 057403 (2002). [CrossRef] [PubMed]

17. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. **82**, 729 (2010). [CrossRef]

28. A. Benabbas, V. Halte, and J. Bigot, “Analytical model of the optical response of periodically structured metallic films,” Opt. Express **13**, 8730–8745 (2005). [CrossRef] [PubMed]

*the simplest analytical condition*which will still be able to predict the emergence of ET,

*based on existing models*. Such a condition has to be able to explain the emergence of ET

*in all of the above configurations*with the same underlying physical mechanism, and be able to give an intuitive understanding of the physics.

10. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. **83**, 2845 (1999). [CrossRef]

12. P. Lalanne, J. Hugonin, S. Astilean, M. Palamaru, and K. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. **2**, 48–51 (2000). [CrossRef]

*effective dielectric medium*, thus greatly simplifying the problem. We will then show that this effective model, which we term the Effective Bragg Cavity (EBC) model, provides excellent agreement with rigorous numerical simulations for various configurations.

## 2. Effective Bragg-cavity model: derivation

*k*,

_{z}*d*is the periodicity of the grating,

*a*the slit width,

*w*is the grating thickness.

*n*

_{1},

*n*

_{3}are the refractive index of the infinite dielectric layers before and after the grating, and

*n*is the refractive index inside the slits. In some of the configurations, an extra thin dielectric layer will be added, with the refractive index

_{s}*n*

_{2}, and thickness

*w*

_{2}that is of the same order of magnitude as the grating thickness

*w*.

28. A. Benabbas, V. Halte, and J. Bigot, “Analytical model of the optical response of periodically structured metallic films,” Opt. Express **13**, 8730–8745 (2005). [CrossRef] [PubMed]

*ɛ*is the dielectric constant and

*μ*is the relative permeability. The vacuum wave vector of the plane wave with a wavelength

*λ*, incident on the grating is

*k*= 2

*π*/

*λ*.

*w*

_{2}= 0 (i.e. no added thin dielectric layer). Assuming a one-dimensional array of slits, periodic along the x axis, with

*μ*(

*r*) = 1 everywhere, the eigenfunctions of Eq. (2) for the magnetic field inside the periodic metal grating will be in the form of Bloch waves, [11

**66** (2002). [CrossRef]

*j*indexes the eigenmode,

*g*= 2

*π*/

*d*,

*k*is the same as that of the incident electromagnetic wave, and

_{x}*ψ*

_{(j)}denotes the excitation of the j-th eigenmode inside the grating. In the dielectric layers 1,3, before and after the grating respectively, the eigenmode solutions are just with

*k*given by

_{z}*k*

_{1,3}=

*k*

_{0}

*n*

_{1,3},

*θ*is the incidence angle depicted in Fig. 1, and

*z*components of the wave vector inside the grating,

27. M. Moharam and T. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A **3**, 1780–1787 (1986). [CrossRef]

*λ*/

*n*> 2

_{s}*a*), there is only one propagating mode inside the slits of the grating. We will denote this mode by

*w*is very large compared to the metal skin depth, one can use the approximation that only the propagating mode is excited by the incoming wave (i.e. discarding the evanescent modes inside the grating) [6

6. F. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B **66** (2002). [CrossRef]

12. P. Lalanne, J. Hugonin, S. Astilean, M. Palamaru, and K. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. **2**, 48–51 (2000). [CrossRef]

29. J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. **94**, 197401 (2005). [CrossRef] [PubMed]

*θ*= 0 in Fig. 1),

*k*= 0 in Eq. (4) and

_{x}*k*=

_{x}*gm*. Solving the boundary conditions between the different layers under the single mode approximation, with the added assumptions that the metal is perfectly conducting leads to the semi-analytical models for the transmission [10

**83**, 2845 (1999). [CrossRef]

**2**, 48–51 (2000). [CrossRef]

*λ*/

*n*

_{1,3}>

*d*(which means

*k*

_{1,3}<

*g*), there can be only one propagating mode outside of the grating, having

*m*≠ 0 are evanescent. However, inside the grating, viewing the excited Bloch mode as a superposition of plane waves with

*m*), all these plane waves are propagating, even those with

*m*≠ 0, as is clearly seen in Eq. (4).

*m*, even though

*k*=

_{x}*gm*≠ 0.

*m*≠ 0). The general condition for a standing wave including all orders of

*m*can be solved analytically with the approximation of a perfectly conducting metal [29

29. J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. **94**, 197401 (2005). [CrossRef] [PubMed]

*a*is of the same order of magnitude as the slit periodicity

*d*, we can with good accuracy take only excited modes with

*m*= ±1. That is because for ideal metals, for the Bloch mode inside the grating in Eq. (4), the relative amplitude

*H*of each order of

_{m}*m*is proportional to the fourier transform of a rectangular box of width

*m*for

*H*using an RCWA numerical calculation for different real metals). In this case, a much simpler picture emerges:

_{m}*we can map this problem into a similar one by replacing the metallic grating with a dielectric material*whose refractive index is defined as For normal incidence light in the TM polarization,

*n*=

*n*, surrounded by two lower refractive index dielectric layers,

_{eff}*n*

_{1},

*n*

_{3}. Thus, solving the standard slab waveguide transverse resonance condition: will give us the values of

*k*which produces the standing wave inside the grating layer for

*m*= 1. This resonant

*k*will be denoted by

*n*given by Eq. (5), and

_{eff}*l*is a non negative integer. For the specific case where

*n*=

_{s}*n*

_{1}=

*n*

_{3}, we get an even simpler form: with

*m*≠ 0) will cause the forward transmission to be at maximum value, because of constructive interference with the propagating

*m*= 0 mode (zero order transmission), similar to a Fabry-Perot cavity. The argument is similar in spirit to the one proposed by Rosenblatt et al. [22

22. D. Rosenblatt, A. Sharon, and A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. **33**, 2038–2059 (1997). [CrossRef]

*in either*TE or TM polarization, the wavelength

*λ*

_{0}for which an ET maxima occurs in rigorous numerical calculations matches well the solution for the analytical equation for the standing wave condition (i.e. Eq. (6)). Therefore, our simplified model is that

*for there to be ET, there has to be a standing wave in the ẑ direction inside the system for the bragg modes having m*≠ 0

*. The ET resonance condition is therefore approximately given by Eq. 6. This, in essence, is the Effective Bragg-Cavity (EBC) Model.*Importantly, this simple model predicts correctly the emergence of ET in a vast variety of 1D configurations, i.e, for both TE and TM polarizations, and for the subwavelength and non-subwavelength spectral regimes,

*using the same analytical condition*(Eq. (6)). The only difference between these different configurations is the wave-number of the propagating mode in grating, and the effective region where this standing wave occurs, as will be explained next. (It is also important to note that while taking higher orders of

*m*into consideration makes it difficult to map the problem to the dielectric picture, for a perfectly conducting metal it is still possible to find a closed form solution using all orders of

*m*, see Ref. [29

29. J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. **94**, 197401 (2005). [CrossRef] [PubMed]

*λ*at an ET maximum in the different configurations. These configurations are summarized in Table. 1. Figure 2(b)(1–3) shows the EBC model mapping, with the standing wave which corresponds to each of the configurations ((

*λ*/

*n*

_{1}), (

*λ*/

*n*

_{1}) <

*d*for all cases):

- corresponds to an incident plane wave in either the TM polarization, in which case both (
*λ*/*n*) ≥ 2_{s}*a*and (*λ*/*n*) < 2_{s}*a*are valid, or in the TE polarization, for*w*_{2}= 0 (no thin dielectric layer) for (*λ*/*n*) < 2_{s}*a*(here both the dielectric materials*n*_{1},*n*_{3}are approximated as having infinite thickness). This is the usual scenario of a bare grating discussed in the literature (specifically, the near field calculation shown coincides with TM polarized light resonance). This standing wave also corresponds to TM polarized incoming light, with an added thin dielectric layer*n*_{2}, for*λ*/*n*_{2}>*d*. - corresponds to the case where a thin dielectric layer
*n*_{2}with a finite thickness is added and*λ*/*n*_{2}<*d*, in both polarizations. In the case of incoming light in the TE polarization, an extra condition*λ*/*n*< 2_{s}*a*applies. - corresponds to an incoming plane wave in the TE polarization, with a thin dielectric layer
*n*_{2}, (*λ*/*n*) > 2_{s}*a*, and (*λ*/*n*_{2}) <*d*.

### 2.1. ET in TE polarization - no thin dielectric layer

*a*in case of non-ideal metal to account for skin depth),

*k*found numerically in rigorous numerical calculations (RCWA) with the one given by Eq. (8). The approximation holds remarkably well as long as the imaginary part of the propagating

_{z}*k*is small, which is valid as long as

_{z}*λ*/

*n*< 2

_{s}*a*. With this approximation of

*w*

_{2}= 0) for the case

*λ*/

*n*< 2

_{s}*a*.

*λ*/

*n*) ≥ 2

_{s}*a*) there are

**no propagating modes**inside the grating. Therefore, the ET in TE polarization, which was both observed in wire grating experiments [26

26. H. Lochbihler, “Enhanced transmission of TE polarized light through wire gratings,” Phys. Rev. B **79** (2009). [CrossRef]

25. D. Crouse and P. Keshavareddy, “Polarization independent enhanced optical transmission in one-dimensional gratings and device applications,” Opt. Express **15**, 1415–1427 (2007). [CrossRef] [PubMed]

26. H. Lochbihler, “Enhanced transmission of TE polarized light through wire gratings,” Phys. Rev. B **79** (2009). [CrossRef]

25. D. Crouse and P. Keshavareddy, “Polarization independent enhanced optical transmission in one-dimensional gratings and device applications,” Opt. Express **15**, 1415–1427 (2007). [CrossRef] [PubMed]

26. H. Lochbihler, “Enhanced transmission of TE polarized light through wire gratings,” Phys. Rev. B **79** (2009). [CrossRef]

### 2.2. ET in TE polarization - with thin dielectric layer

*λ*/

*n*) > 2

_{s}*a*), because of the lack of a propagating wave inside the grating, as seen from the cutoff of

*λ*/

*n*> 2

_{s}*a*(given that

*λ*/

*n*

_{2}<

*d*) [21

21. E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Extraordinary optical transmission without plasmons: the s-polarization case,” J. Opt. A: Pure Appl. Opt. **8**, S94–S97 (2006). [CrossRef]

30. M. Guillaumee, A. Y. Nikitin, M. J. K. Klein, L. A. Dunbar, V. Spassov, R. Eckert, L. Martin-Moreno, F. J. Garcia-Vidal, and R. P. Stanley, “Observation of enhanced transmission for s-polarized light through a subwavelength slit,” Opt. Express **18**, 9722–9727 (2010). [CrossRef] [PubMed]

*n*

_{2}) can still support one, allowing for a standing wave in the thin dielectric layer. In this case, the grating acts as one of the boundaries.

*m*= 1) for both the grating and the thin dielectric layer

*n*

_{2}((

*λ*/

*n*) < 2

_{s}*a*), the standing wave will be in both these layers, and is given by the equation for a two layer dielectric waveguide (with

*n*

_{2}and

*n*for the grating layer), corresponding to Fig. 2(b)(2).

_{eff}*λ*/

*n*)

_{s}*>*2

*a*), the thin dielectric layer can still support a propagating mode for the first bragg order (given that

*λ*/

*n*

_{2}<

*d*). Even though there is no propagating mode in the grating, there will still be an evanescent eigenmode with a relatively small imaginary wave vector (as long as (

*λ*/

*n*) is only slightly larger than 2

_{s}*a*), which will be denoted by

21. E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Extraordinary optical transmission without plasmons: the s-polarization case,” J. Opt. A: Pure Appl. Opt. **8**, S94–S97 (2006). [CrossRef]

*n*

_{2}: mapping the grating into a homogeneous dielectric layer lets us use Eq. (6) for finding the standing wave condition, with the changes

### 2.3. ET in TM polarization - with thin dielectric layer

*m*= 1) is evanescent in all the homogeneous dielectric layers (

*λ*/

*n*>

_{i}*d*, for

*i*∈ {1,2,3}). In this case the ET corresponds to a standing wave inside the grating layer, with layers

*n*

_{1},

*n*

_{2}acting as the boundaries. This is given by Eq. (6), with

*n*

_{2}is very thin, or where the the mode is only slightly evanescent in the thin dielectric layer, one needs to take into account also the layer

*n*

_{1}, leading to a slightly more complicated standing wave condition. This case corresponds to Fig. 2(b)(1).

*m*= 1) is propagating in the thin dielectric layer

*n*

_{2}, but evanescent in layers 1,3. In this case, as in the non-subwavelengh regime for incoming light in the TE polarization with an extra thin dielectric layer (section 2.2), the ET will correspond to standing wave in both the grating and the thin dielectric layer. This will be given by the equation for a two layer dielectric waveguide (with

*n*

_{2}for the thin dielectric layer and

*n*for the grating layer), and correspond to Fig. 2(b)(2).

_{eff}## 3. Comparison to numerical calculations

**66** (2002). [CrossRef]

27. M. Moharam and T. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A **3**, 1780–1787 (1986). [CrossRef]

### 3.1. TM polarization

*d*= 0.9

*μm*,

*a*= 0.35

*μm*, and (

*n*

_{1}=

*n*

_{3}=

*n*= 1) on both sides of the grating and inside the slits. The predictions of the ET maxima, given by Eq. (6) are plotted in Fig. 3(a), by the white dotted lines. A very good agreement between the numerically calculated transmission maxima and the EBC model is clearly seen, with no fitting parameters.

_{s}**66** (2002). [CrossRef]

*l*being an integer), similar to Fabry-Perot cavities. This region is termed the slit cavity-like ET (which also exhibits a high intensity of the electromagnetic field inside the slits). There is also a smooth transition between this region and the region marked by (2) in Fig. 3(a), which deviates from the linear slope. This region is termed SPP-like ET (with the local field intensities centralized inside the slits and outside them as well). As seen, this transition is also predicted by the EBC model. Indeed, since in this configuration we have

*n*

_{1}=

*n*

_{3}=

*n*, in the limit where

_{s}*n*≫

_{eff}*n*

_{1,3}and

*χ*≫ 1. Therefore, in this limit, Eq. (7) becomes

*ϕ*

_{12}=

*ϕ*

_{23}≈

*tan*

^{−1}(

*χ*

^{3}) ≈

*π*/2 and Eq. (6) reduces to 2

*n*

_{s}k_{0}

*w*= 2

*πl*which is the pure metallic slab waveguide condition, or with

*l*∈ ℕ, as in a typical Fabry-Perot resonance in the metallic grating. This means that the standing wave is confined exactly inside the metallic grating, corresponding to the slit cavity-like ET maxima in Ref. [10

**83**, 2845 (1999). [CrossRef]

*ẑ*direction which is much larger than the grating width

*w*. This limit corresponds to the SPP-like ET in Ref. [10

**83**, 2845 (1999). [CrossRef]

### 3.2. TE polarization - no thin dielectric layer

*d*= 0.9

*μm*and

*a*= 0.55

*μm*. we can see that for (

*λ*/

*n*) <

_{s}*d*the transmission is not dependent on the grating thickness, and no ET resonance is observed. However, since

*d*< 2

*a*we see ET in this polarization when

*d*< (

*λ*/

*n*) < 2

_{s}*a*, again in good agreement with the EBC model. As can be seen, the behavior of the ET lines differs from the TM polarization. This difference is largely explained by the difference in the metallic slab waveguide equations (and thus

### 3.3. TE polarization - with thin dielectric layer

*d*= 0.9

*μm*,

*a*= 0.35

*μm*,

*w*

_{2}= 0.93

*μm*,

*n*

_{1}=

*n*

_{3}= 1 and

*n*

_{2}= 1.52.

*λ*/

*n*) < 2

_{s}*a*), corresponding to the region beneath the black dashed line in Fig. 5, the maximum transmission lines behave similarly to the configuration without an added dielectric layer (see Fig. 4 for comparison and the discussion in Sec. 3.2). However, the effective thickness is different - the cavity in this case is the grating layer + dielectric layer

*n*

_{2}. It can also be seen that ET is observed in the subwavelength regime (above the dashed black line) as predicted by the EBC model in Sec. 2.2. This is due to the evanescent coupling discussed earlier. The extra observed features of transmission minima lines closely correspond to the waveguide condition in the thin dielectric layer

*n*

_{2}, taking the grating as a homogeneous metallic slab [21

21. E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Extraordinary optical transmission without plasmons: the s-polarization case,” J. Opt. A: Pure Appl. Opt. **8**, S94–S97 (2006). [CrossRef]

*a*, we can change the properties of the evanescent wave in the slits. This change is manifested by a change in

*a*is changed. In Fig. 6 the zero order transmission maxima is extracted from the numerical model for different values of the slit width

*a*in the subwavelength regime, and is compared to the predicted value given by the EBC model (the full RCWA transmission map makes the maximum hard to see, and so is not included). As can be seen, there is a good correspondence between the two. Note that the whole wavelength spectral range shown in Fig. 6 is only 20

*nm*, so we do not expect a perfect fit on this spectral scale, as was explained before. However it is clear that the trend of both lines is the same.

**94**, 197401 (2005). [CrossRef] [PubMed]

**2**, 48–51 (2000). [CrossRef]

**94**, 197401 (2005). [CrossRef] [PubMed]

*n*

_{1}≠

*n*

_{3}. In the spectral regimes where all the higher bragg diffraction orders are evanescent in both infinite dielectric layers (layers 1,3), the agreement with our analytical model was just as good. When one of the dielectric layers starts supporting a propagating mode with

*m*≠ 0, no real ET is apparent as is expected.

## 4. Conclusions

*a*and

*d*are of the same order of magnitude, allowed a derivation of a simple analytical closed form condition (Eq. (6)) for the spectral position of the ET maxima and of the EBC model: a mapping of the problem to a much simpler one of a standing wave condition with an effective homogeneous dielectric layer instead of the metal grating. We have analyzed various different configuration in which ET emerges, and explained them all as being realizations of the same standing wave model, with the standing wave being confined in different layers for different configurations. Finally, we compared the model predictions with rigorous numerical calculations and have shown a very good agreement between the two.

1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**, 667–669 (1998). [CrossRef]

## Acknowledgments

## References and links

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5. | N. Livneh, A. Strauss, I. Schwarz, I. Rosenberg, A. Zimran, S. Yochelis, G. Chen, U. Banin, Y. Paltiel, and R. Rapaport, “Highly directional emission and photon beaming from nanocrystal quantum dots embedded in metallic nanoslit arrays,” Nano Lett. |

6. | F. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B |

7. | M. G. Harats, I. Schwarz, A. Zimran, U. Banin, G. Chen, and R. Rapaport, “Enhancement of two photon processes in quantum dots embedded in subwavelength metallic gratings,” Opt. Express |

8. | X. Zhang, H. Liu, J. Tian, Y. Song, and L. Wang, “Band-Selective optical polarizer based on Gold-Nanowire plasmonic diffraction gratings,” Nano Lett. |

9. | F. Chien, C. Lin, J. Yih, K. Lee, C. Chang, P. Wei, C. Sun, and S. Chen, “Coupled waveguide-surface plasmon resonance biosensor with subwavelength grating,” Biosens. Bioelectron. |

10. | J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. |

11. | M. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B |

12. | P. Lalanne, J. Hugonin, S. Astilean, M. Palamaru, and K. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A: Pure Appl. Opt. |

13. | J. Shen and P. Platzman, “Properties of a one-dimensional metallophotonic crystal,” Phys. Rev. B |

14. | K. G. Lee and Q. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. |

15. | Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. |

16. | J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science |

17. | F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. |

18. | A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, “Resonant transmittance through metal films with fabricated and light-induced modulation,” Phys. Rev. B |

19. | S. A. Darmanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: An analytical study,” Phys. Rev. B |

20. | A. V. Kats and A. Y. Nikitin, “Analytical treatment of anomalous transparency of a modulated metal film due to surface plasmon-polariton excitation,” Phys. Rev. B |

21. | E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Extraordinary optical transmission without plasmons: the s-polarization case,” J. Opt. A: Pure Appl. Opt. |

22. | D. Rosenblatt, A. Sharon, and A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. |

23. | P. S. Priambodo, T. A. Maldonado, and R. Magnusson, “Fabrication and characterization of high-quality waveguide-mode resonant optical filters,” Appl. Phys. Lett. |

24. | Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express |

25. | D. Crouse and P. Keshavareddy, “Polarization independent enhanced optical transmission in one-dimensional gratings and device applications,” Opt. Express |

26. | H. Lochbihler, “Enhanced transmission of TE polarized light through wire gratings,” Phys. Rev. B |

27. | M. Moharam and T. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A |

28. | A. Benabbas, V. Halte, and J. Bigot, “Analytical model of the optical response of periodically structured metallic films,” Opt. Express |

29. | J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. |

30. | M. Guillaumee, A. Y. Nikitin, M. J. K. Klein, L. A. Dunbar, V. Spassov, R. Eckert, L. Martin-Moreno, F. J. Garcia-Vidal, and R. P. Stanley, “Observation of enhanced transmission for s-polarized light through a subwavelength slit,” Opt. Express |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.2065) Diffraction and gratings : Effective medium theory

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: September 6, 2011

Revised Manuscript: November 6, 2011

Manuscript Accepted: November 11, 2011

Published: December 21, 2011

**Citation**

Ilai Schwarz, Nitzan Livneh, and Ronen Rapaport, "General closed-form condition for enhanced transmission in subwavelength metallic gratings in both TE and TM polarizations," Opt. Express **20**, 426-439 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-426

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### References

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- F. Chien, C. Lin, J. Yih, K. Lee, C. Chang, P. Wei, C. Sun, and S. Chen, “Coupled waveguide-surface plasmon resonance biosensor with subwavelength grating,” Biosens. Bioelectron.22, 2737–2742 (2007). [CrossRef]
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